Loose coupling method between a stess analysis system and a conventional reservoir simulator

ABSTRACT

Methods for loosely coupling a stress analysis system to a conventional reservoir simulator by adjusting the flow equation of the conventional reservoir simulator. The solution is obtained by using the methods in a loose, iterative coupling system such than when convergence is reached, the results obtained are close to those of the full coupling system. A system for implementing the methods on a digitally readable medium.

STATEMENT OF RELATED APPLICATIONS

This patent application is the US National Stage under 35 USC 371 ofPatent Cooperation Treaty (PCT) International Application No.PCT/BR2010/000096 having an International Filing Date of 31 Mar. 2010,which claims priority on Brazilian Patent Application No. PI 0900908-6having a filing date of 31 Mar. 2009.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention is directed to methods for loosely coupling astress analysis system (geomechanical simulator) and a conventionalreservoir simulator. In particular, the methods of the invention providethe adjusting of the flow equation of the conventional reservoirsimulator, considering the variation of the stress state of the rock.Therefore, the solution obtained using the methods according to theinvention in a loose, iterative coupling system, when convergence isreached the results obtained by this solution are close to those of thefull coupling system. The present invention is also directed to a systemfor implementing the invention provided on a digitally readable medium.

2. Prior Art

Recently, there has been great the interest in the study of loosecoupling between stress analysis systems and conventional reservoirsimulators. The problem of flow and stress can be coupled by usingdifferent coupling systems, the three main systems are: full coupling,loose iterative coupling and loose explicit coupling.

In full coupling, the set of equations governing the hydro-mechanicalproblem is solved simultaneously by a single simulator, featuring themore rigorous formulation of coupling.

In the loose iterative coupling (or two-way coupling), the flow andstress equations are solved separately and sequentially for each timeinterval. FIG. 1 (a) illustrates this system, the information areexchanged in the same time interval, between the conventional reservoirsimulator and the stress analysis system until it reaches theconvergence of an unknown variable, for example, pressure.

In loose explicit coupling (or one-way coupling), only the conventionalreservoir simulator sends information (variation of pore pressure) tothe stress analysis system. FIG. 1( b) illustrates this system, noinformation is sent from the stress analysis system to the conventionalreservoir simulator, so the flow problem is not affected by changing instress state in the reservoir and surrounding rocks.

The four main studies published in the literature concerning loosecoupling between stress analysis systems and conventional reservoirsimulators are briefly described below.

Settari and Mounts¹ presented one of the first studies about loosecoupling of a conventional reservoir simulator (DRS-STEAM) and a stressanalysis system (FEM3D). The authors presented an algorithm for looseiterative coupling and the porosity, depending on the pore pressurefield and the stress state, which was the only coupling parameter used.

Mainguy and Longuemare⁴ had presented three equations to correct theequation for porosity frequently used in conventional reservoirsimulator to take into account the variation in stress state. Again, theporosity was the only coupling parameter used in the loose coupling.

Dean et al.³ showed results of three hydro-mechanical coupling system:loose explicit, loose iterative and full. In the coupling system calledby the authors as loose explicit, the porosity is evaluated in twoways: 1) considering the variation of the pore pressure field and stressstate; 2) considering the variation of pore pressure field and thecalculation of a new compressibility, evaluated by the simple loadinghypothesis, such as uniaxial deformation. Only the porosity wasconsidered as coupling parameter in both loose coupling systems. Thethree coupling systems were implemented in the program Acres (ARCOS'sComprehensive Reservoir Simulator).

Samier and Gennaro² proposed a new loose iterative coupling system. Inthis new coupling system, the iterations are not performed in timeintervals, they are performed over the complete time of analysis. Theeffect of stress analysis is introduced in the porosity through afunctionality of the conventional reservoir simulator called pore volumemultiplier. Again, the porosity was the only coupling parameterconsidered.

From reading these four studies, it can be concluded that only theporosity is used as coupling parameter between the stress analysissystem and the conventional reservoir simulator.

The Governing Equations

In the present invention, the governing equations are formulated usingcontinuum mechanics, which analyzes the mechanical behavior of modeledmaterials as a continuous (solid, liquid and gases).

The Governing Equations for the Flow Problem

The flow equation is obtained from the mass conservation law. The massconservation law can be represented mathematically by the equation:

$\begin{matrix}{{{- \nabla} \cdot ( {\rho_{f}v} )} = {\frac{\partial\;}{\partial t}( {\rho_{f}\varphi} )}} & (1)\end{matrix}$

wherein v is the velocity vector (L/t), ρ_(f) is the specific mass ofthe fluid (m/L³) and φ is the porosity.

Darcy's law states that the rate of percolation is directly proportionalto the pressure gradient:

$\begin{matrix}{v = {{- \frac{k}{\mu}}{\nabla p}}} & (2)\end{matrix}$

wherein k is the absolute permeability (L²), μ is the viscosity (m/Lt)and p the pore pressure (m/Lt²).

In geomechanical terms, the difference in the development of the flowequation used in conventional reservoir simulation and in the fullcoupling system is in development of the formulation adopted to evaluatethe porosity. In conventional reservoir simulation, the variation ofporosity can be related to the variation of pore pressure through thecompressibility of rock, using a linear relation.

φ=φ⁰[1+c _(r)(p−p ⁰)]  (3)

wherein φ⁰ is the inicial porosity, p⁰ is the initial pore pressure andc_(r) is the compressibility of the rock that can be calculated as:

$\begin{matrix}{c_{r} = {{\frac{1}{\varphi_{0}}\frac{\partial\varphi}{\partial p}} = {\frac{1}{V_{p}^{0}}\frac{\partial V_{p}}{\partial p}}}} & (4)\end{matrix}$

wherein V_(p) ⁰ is the porous volume in the initial configuration andV_(p) is the porous volume in the final configuration.

The fluid compressibility is taken into account in conventionalreservoir simulation using the equation:

ρ_(f)=ρ_(f) ⁰[1+c _(f)(p−p ⁰)]  (5)

wherein ρ⁰ _(f) is the initial specific mass of the fluid and c_(f) isthe fluid compressibility which relates the variation of specific masswith the variation of pore pressure.

$\begin{matrix}{c_{f} = {{\frac{1}{\rho_{f}^{0}}\frac{\partial\rho_{f}}{\partial p}} = \frac{1}{K_{f}}}} & (6)\end{matrix}$

wherein K_(f) is the modulus of volumetric deformation of the fluid.

Introducing equation (3) and (5) in equation (1), the variation ofporosity and specific mass of the fluid over time (accumulation term)can be considered in the simulation of conventional reservoirs. Thefinal form of the flow equation can be written as:

$\begin{matrix}{{{( {{c_{f}\varphi^{0}} + {c_{r}\varphi^{0}}} )\frac{\partial p}{\partial t}} - {\frac{k}{\mu}{\nabla^{2}p}}} = 0} & (7)\end{matrix}$

To an elastic, linear and isotropic analysis, the expression of thevariation of porosity used in the full coupling system of is comprisedof four components that contribute to fluid accumulation (Zienkiewicz etal.⁵).

a) The variation of volumetric deformation: −dε_(v);

b) The variation due to compression of the solid matrix by porepressure: (1−n) dp/K_(S);

c) The variation due to compression of the solid matrix by effectivestress: −K_(D)/K_(S) (dε_(v)+dp/Ks);

d) The variation due to compression of the fluid by pore pressure:ndp/K_(f).

The equation of the variation of porosity is obtained from the sum ofthe above four components:

$\begin{matrix}{\varphi = {\varphi^{0} + {\alpha ( {ɛ_{v} - ɛ_{v}^{0}} )} + {\frac{1}{Q}( {p - p^{0}} )}}} & (8)\end{matrix}$

wherein ε⁰ _(v) is the initial volumetric deformation (solids+pores) andε_(v) is the final volumetric deformation (solids+pores). The parameterQ of Biot⁶ is written as:

$\begin{matrix}{\frac{1}{Q} = {{\frac{\varphi^{0}}{K_{f}} + \frac{\alpha - \varphi^{0}}{K_{S}}} = {{c_{f}\varphi^{0}} + {c_{S}( {\alpha - \varphi^{0}} )}}}} & (9)\end{matrix}$

wherein c_(s) is the compressibility of solid matrix (c_(s)=1/K_(s)) ansK_(s) is the modulum of volumetric deformation of solid matrix.

The parameter a of Biot⁶ is written in terms of the module of volumetricdeformation of the rock and the pores K_(D) and the volumetricdeformation modulus of solid matrix K_(S) (Zienkiewics et al⁵).

$\begin{matrix}{\alpha = {1 - \frac{K_{D}}{K_{S}}}} & (10)\end{matrix}$

wherein K_(D) is the volumetric deformation modulus associated to thedrained constitutive matrix tangent C.

$\begin{matrix}{K_{D} = {\frac{m^{T}{Cm}}{9} = \frac{E}{3( {1 - {2v}} )}}} & (11)\end{matrix}$

wherein m is the identity matrix, E is the Young modulus and v is thePoisson's ratio.

Insert equations (5) and (8) in equation (1), the variation of porositythat takes into account the variation of stress state is considered interms of accumulation, resulting in the flow equation of full couplingsystem.

$\begin{matrix}{{{\lbrack {{c_{f}\varphi^{0}} + {c_{S}( {\alpha - \varphi^{0}} )}} \rbrack \frac{\partial p}{\partial t}} - {\frac{k}{\mu}{\nabla^{2}p}}} = {{- \alpha}\frac{\partial ɛ_{v}}{\partial t}}} & (12)\end{matrix}$

The Governing Equations for the Problem Geomechanical

The balance equation is written as:

∇·σ+ρb=0  (13)

wherein δ is the total stress tensor and ρ is the total density of thecomposition, that is:

ρ=φρ_(f)+(1−φ)ρ_(S)  (14)

wherein ρ_(s) is the density of solid matrix.

The tensor of deformations ε can be written in terms of the vectordisplacement u as:

ε=½[∇u+(∇u)^(T)]  (15)

The principle of Terzaghi effective stress is written as:

σ′=σ−αmp  (16)

wherein δ′ is the effective stress tensor.

The effective stress tensor is related to the tensor of deformationsthrough the drained constitutive tangent c matrix.

σ′=C:ε  (17)

Introducing equations (15), (16) and (17) in equation (13), the balanceequation coupled in terms of displacement and pore pressure can bewritten as:

$\begin{matrix}{{{G{\nabla^{2}u}} + {\frac{G}{1 - {2v}}{{\nabla\nabla} \cdot u}}} = {\alpha {\nabla p}}} & (18)\end{matrix}$

wherein G is the shear module.

The Governing Equations for the Partial Coupling Scheme

FIG. 2 shows the assembly of the governing equations of the loosecoupling system, the flow equation (7) is obtained from the conventionalreservoirs simulation and the mechanical behavior is governed by thebalance equation (12) written in terms of displacement and porepressure, the same used in the full coupling system. The challenge ofthis coupled problem is to get from flow equation (7) of conventionalreservoir simulation the same response of flow equation (12) of the fullcoupling system. In general, conventional reservoir simulators areclosed-source software (proprietary software), hampering the process ofloose coupling, so it is necessary to use external artifices to reshapethe flow equation (7) conventional reservoir simulator.

Comparing the flow equations (7) and (12) can be observed that the termsc_(f)φ⁰∂p/∂t are common in the equations. The term c_(r)φ⁰∂p/∂t is foundonly in the equation (7) and the terms c_(s)(α−φ⁰)∂p/∂t and α∂ε_(v)/∂tare found only in equation (12).

The literature references cited above do not anticipate or even suggestthe invention scope. They are listed below in more detail for simpleverification.

-   (1) A. Settari and F. M. Mourits, “Coupling of Geomechanics and    Reservoir Simulation Models”, Computer Methods and Advances in    Geomechanics, Siriwardane & Zanan (Eds), Balkema, Rotterdam (1994).-   (2) P. Samiei and S. De Gennaro, “Practical Interative Coupling of    Geomechanics with Reservoir Simulation”, SPE paper 106188 (2007).-   (3) R. H. Dean, X. Gai, C. M. Stone, and S. Mikoff, “A Comparison of    Techniques for Coupling Porous Flow and Geomechanics”, paper SPE    79709 (2006).-   (4) M. Mainguy and P. Longuemare, “Coupling Fluid Flow and Rock    Mechanics: Formulations of the Partial Coupling Between    Geomechanical and Reservoir Simulators”, Oil & Gas Science and    Technology, Vol 57, No. 4, 355-367 (2002).-   (5) O. C. Zienkiewicz, A. H. C. Chan, M. Shepherd, B. A. Schrefler,    and T. Shiomi, “Computational Geomechanics with Special Reference to    Earthquake Engineering”, John Wiley and Sons, (1999).-   (6) M. A. Biot, “General Theory of Three-Dimensional    Consolidation”, J. Appl. Phys., Vol 12, 155-164, (1940).

Prior art patent has documents relating to this topic, in which the mostsignificant are described below.

-   Document U.S. Pat. No. 7,386,431 describes a system and method for    modeling and simulating the event of “fracture” in oil wells,    especially the phenomena known as “interfacial slip” or “debonding”    between adjacent layers of the Earth formation.-   Document U.S. Pat. No. 7,177,764 describes a method to calculate    stress around a fault while computing the prediction of the rock    stress/fluid flow by means of the conservation of momentum. The    method takes into account the presence of a multi-phase flow, where    the number of flowing fluid phases is between 1 and 3.-   Document WO 2008/070526 describes a method for calculation and    simulating fluid flow in a fractured subterranean reservoir from the    combination of discrete fracture networks and homogenization of    small fractures.

The present invention differs from these documents by providingalternatives that allow approximating the flow equation of theconventional reservoir simulation to the flow equation of the fullcoupling system, removing the rock compressibility effect (c_(r)φ⁰∂p/∂t)and adding the volumetric deformation effect of rock and the pores(α∂ε_(v)/∂t).

Therefore, can be observed that none of the mentioned documents discloseor even suggest at the concepts of the present invention, so that itpresents the requirements for patentability.

BRIEF SUMMARY OF THE INVENTION

In a first aspect, is one of the objects of the invention to providesolutions to approximate the flow equation of the conventional reservoirsimulation to the flow equation of the full coupling system, consideringthe effect of the variation of stress state in the reservoir simulation.The present invention therefore provides a loose iterative couplingsystem between a stress analysis system and a conventional reservoirsimulator to obtain similar responses from those obtained fromsimulators that use a full coupling system, achieved the convergence ofthe iterative system.

It is another aspect, the invention provides the novel feature ofremoving the rock compressibility effect (c_(r)φ⁰∂p/∂t) and adding thevolumetric deformation effect of rock and the pores (α∂ε_(v)/∂t) inorder to approximate the flow equation of the conventional reservoirsimulation to the flow equation of the full coupling system.

A further aspect of the present invention provides a system capable ofperforming the method of the invention, wherein said system is containedin a digitally readable medium.

These and other objects of the present invention will be betterunderstood based on the detailed description below.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows two loose coupling systems: (a) Iterative and (b) Explicit;where n is the number of time intervals; n_(g) is the number ofiterations, 1.1 represents the Conventional Reservoir Simulator; 1.2 theGeomechanical Simulator; 1.3 the Culpling Parameters; and 1.4 theConvergence.

FIG. 2 shows the assembly of the governing equations of the loosecoupling system, where 2.1 represents the Conventional ReservoirsSimulation, 2.2 Full Coupling System, and 2.3 the Loose Coupling System.

FIG. 3 shows the boundary conditions imposed by natural and essentialboundary r.

FIG. 4 shows methodology A—Approximation of the flow equation of theconventional reservoir simulator through the addition/removal of rate offluid and porosity, which 1.1 represents Conventional ReservoirSimulator, 1.2 the Geomechanical Simulator; 1.4 the Convergence, 3.1 theNodal Forces, 3.2 Fluid Rate Add/Removed and porosity, and ** theUnknowns.

FIG. 5 shows methodology B—Approximation of the flow equation of theconventional reservoir simulator through the pseudo-compressibility ofthe rock and porosity, where 1.1 represents the Conventional ReservoirSimulator, 1.2 the Geomechanical Simulator; 1.4 the Convergence, 3.1 theNodal Forces; 4.1 the Pseudo-compressibility of the Rock and porosity,and *** the Unknowns.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

In this section will presented the steps involved in the method of theinvention, which can be embodied through two similar approaches toapproximate the flow equation of the conventional reservoirs simulationof the flow equation of the full coupling system.

The method of the invention consists of the loose coupling methodbetween a stress analysis system to a conventional reservoir simulator,by adjusting the flow equation of the conventional reservoir simulatorthrough:

(i) removing the effect of compressibility of rock (c_(r)φ⁰∂p/∂t) and(ii) the addition of the effect of volumetric deformation of the rockand the pores (α∂ε_(v)/∂t).

This method provides a solution from a loose iterative coupling systemuntil obtain convergence.

In a preferred embodiment, the method of the invention comprisesadjusting the flow equation of the conventional reservoir simulator, sothat it is similar to the flow equation of full coupling by the additionor removal of flow rate using wells (known for the purposes of thepresent invention, as the methodology A).

Methodology A: The first methodology is the use of production wells andinjection wells in each cell of the grid of simulation to remove theinfinitesimal of flow rate (c_(r)φ⁰∂p/∂t) or add infinitesimal rate offluid (α∂ε_(v)/∂t). The well data are written in the input file of theconventional reservoirs simulator.

To calculate the correct values of the flow rate is necessary to use theapproximate solution by finite differences of the partial differentialequation of flow. To simplify the development of the formulation, thefinite difference approximation will be employed in a single phase andone-dimensional problem, according to equation (19).

$\begin{matrix}{{{T_{i + \frac{1}{2}}^{n}p_{i + 1}^{n + 1}} - {\lbrack {( \frac{V_{b}\varphi \; c_{f}}{a_{c}B^{0}\Delta \; t} )_{i} + ( \frac{V_{b}\varphi^{0}c_{r}}{a_{c}B\; \Delta \; t} )_{i} + T_{i + \frac{1}{2}}^{n} + T_{i - \frac{1}{2}}^{n}} \rbrack p_{i}^{n + 1}} + {T_{i - \frac{1}{2}}^{n}p_{i - 1}^{n + 1}}} = {{- \lbrack {( \frac{V_{b}\varphi \; c_{f}}{a_{c}B^{0}\Delta \; t} )_{i} + ( \frac{V_{b}\varphi^{0}\; c_{r}}{a_{c}B\; \Delta \; t} )_{i}} \rbrack}p_{i}^{n}}} & (19)\end{matrix}$

where T is the transmissibility, V_(b) is the volume of rock(solid+pores) (L³), B is the formation volume factor (L³/L³), Δ_(t) isthe time interval (t), n is the time interval previous n+1 is thecurrent time interval and ⁰ is the instant reference.

The first underlined term in equation (20) represents the increment ofrate of fluid removed due to the effect of compressibility of the rockand the second underlined term is the increment of rate of fluid addeddue to the effect of volumetric deformation of the rock (solid+pores).The rock matrix is considered incompressible in the development of thisformulation (c_(s)=O).

$\begin{matrix}{{{{{T_{i + \frac{1}{2}}^{n}p_{i + 1}^{n + 1}} - {\lbrack {( \frac{V_{b}\varphi \; c_{f}}{a_{c}B^{0}\Delta \; t} )_{i} + ( \frac{V_{b}\varphi^{0}c_{r}}{a_{c}B\; \Delta \; t} )_{i} + T_{i + \frac{1}{2}}^{n} + T_{i - \frac{1}{2}}^{n}} \rbrack p_{i}^{n + 1}} + {T_{i - \frac{1}{2}}^{n}p_{i - 1}^{n + 1}}} = {{{- \lbrack {( \frac{V_{b}\varphi \; c_{f}}{\alpha_{c}B^{0}\Delta \; t} ) + ( \frac{V_{b}\varphi^{0}c_{r}}{\alpha_{c}B^{0}\Delta \; t} )} \rbrack}p_{i}^{n}} - \underset{\_}{( \frac{V_{b}\varphi^{0}c_{r}}{\alpha_{c}B^{0}\Delta \; t} )( {p_{i}^{n - 1} - p_{i}^{n}} )} + \underset{\_}{\frac{V_{b}{\Delta ɛ}_{V}}{\alpha_{c}B^{0}\Delta \; t}}}}\mspace{79mu} {{where}\text{:}}}\mspace{40mu}} & (20) \\{ \mspace{79mu} i )\mspace{14mu}} & \; \\{\mspace{79mu} {{{{if}\mspace{14mu} ( \frac{V_{b}\varphi^{0}c_{r}}{\alpha_{c}B^{0}\Delta \; t} )( {p_{i}^{n - 1} - p_{i}^{n}} )} + \frac{V_{b}{\Delta ɛ}_{V}}{\alpha_{c}B^{0}\Delta \; t}} > 0}} & (21)\end{matrix}$

injection wells should be used; and

$\begin{matrix}{ {ii} )\mspace{11mu}} & \; \\{{{{if}\mspace{14mu} ( \frac{V_{b}\varphi^{0}c_{r}}{\alpha_{c}B^{0}\Delta \; t} )( {p_{i}^{n - 1} - p_{i}^{n}} )} + \frac{V_{b}\Delta \; ɛ_{V}}{\alpha_{c}B^{0}\Delta \; t}} < 0} & (22)\end{matrix}$

production wells should be used.

In another preferred embodiment, the method of the invention comprisesadjusting the flow equation of the conventional reservoir simulator, sothat it is similar to the flow equation of full coupling system, byintroducing a pseudo-compressibility (referred to purposes of thepresent invention, as methodology B).

Methodology B: In the second methodology, the compressibility of therock (c_(r)) is used as the coupling parameter. The compressibility ofthe rock calculated should ensure that the response of the flow equationof the conventional reservoir simulation is the same or approximate ofthe flow equation of full coupling system. This compressibility will becalled pseudo-compressibility of the rock and can be evaluated as:

$\begin{matrix}{c_{r_{pseudo}} = \frac{ɛ_{V_{i}}^{n + 1} - ɛ_{V_{i}}^{n}}{\varphi^{n + 1}( {p_{i}^{n - 1} - p_{i}^{n}} )}} & (25)\end{matrix}$

The pseudo-compressibility of the rock, the porosity and pore pressurecalculated at the end of the time interval should be rewritten in theinput file of the conventional reservoir simulator. The porositycalculated at the end of time interval should be introduced with aporosity of reference (φ⁰). If the analysis is performed using the looseiterative coupling system, the pore pressure reference (p⁰) in equation(3) should be calculated pore pressure at the end of time interval. Bythe time the analysis converge p^(n+1)=p^(n)eφ^(n+1)=φ^(n), isguaranteed a unique and consistent solution with the full couplingsystem.

The method of the present invention may be embodied in a computerreadable medium containing programation code means and means to performsuch method.

Below are described two streams of work where the two methodologies ofloose coupling between a stress analysis system and a conventionalreservoir simulator, are used within a loose iterative coupling system,providing an example of application of the methodologies. The followingworkflows are intended only to illustrate the various ways of embodimentof the present invention. Should therefore be seen as an illustration,and not restriction, so that achievements here not described, but in thespirit of the invention are protected by this.

Example 1 Application of Two Methodologies of Loose Coupling Between aStress Analysis System and a Conventional Reservoir Simulator within aTime Interval in Loose Iterative Coupling System

FIG. 4 illustrates the workflow for using the Methodology A within aloose iterative coupling system, considering a time interval. Thepartial coupling system was divided into four steps, which are describedbelow. In step 1 the conventional reservoir simulator calculates theprimary variables of reservoir simulation during the time intervalconsidered: field of pressure (p), field of saturation (S) and field oftemperature (T). In step 2 the variation of the field pore pressure inthe time interval is used to calculate the nodal forces to be applied onthe nodes of the finite element mesh. In step 3 the stress analysissystem calculates the field of displacement (u), the state ofdeformation (ε) and the stress state (σ) resulting from the applicationof nodal forces. In step 4 the parameters to approximate the flowequation of the conventional reservoir simulation of flow equation ofthe full coupling system are calculated using equations (21) and (22).If the convergence of the iterative system is not reached, theaddition/removal of flow rate and the new field of porosity are used ina new reservoir simulation at the same time interval.

FIG. 5 illustrates the workflow for using the methodology B, within aloose iterative coupling system, considering a time interval. Thepartial coupling scheme was also divided into four stages, but only step4 is different from the workflow described above. In this methodology iscalculated a pseudo-compressibility of the rock through the equation(25), which approximates the flow equation of the conventional reservoirsimulation of the flow equation of full coupling system. Again, ifconvergence is not reached, the pseudo-compressibility of the rock andthe new field of porosity are used in a new reservoir simulation at thesame time interval.

Those skilled in the art will immediately value the knowledge hereinshown, know that small variations in the way of realizing theillustrative examples provided here should be considered within thescope of the invention and the attached claims.

1. A loose coupling method between a stress analysis system and aconventional reservoir simulator, comprising adjusting the flow equationof the conventional reservoir simulator through: (i) removing the rockcompressibility effect (c_(r)φ⁰δp/δt); and (ii) adding the volumetricdeformation effect of rock and the pores (α∂ε_(v)/∂t), said methodproviding a solution from a loose iterative coupling system until obtainobtaining the convergence.
 2. The method according to claim 1, whereinthe compressibility of the solid matrix is null.
 3. The method accordingto claim 1, wherein the said adjustment of the flow equation of theconventional reservoir simulator, so that this is similar to the flowequation of the full coupling system, is conducted through the additionor removal of flow rate using wells, according to equations i) or ii):$\begin{matrix} i ) & \; \\{{{{{if}\mspace{14mu} ( \frac{V_{b}\varphi^{0}c_{r}}{\alpha_{c}B^{0}\Delta \; t} )( {p_{i}^{n - 1} - p_{i}^{n}} )} + \frac{V_{b}{\Delta ɛ}_{V}}{\alpha_{c}B^{0}\Delta \; t}} > 0}{{{injection}\mspace{14mu} {wells}\mspace{14mu} {should}\mspace{14mu} {be}\mspace{14mu} {used}};{and}}} & \; \\ {ii} ) & \; \\{{{{if}\mspace{14mu} ( \frac{V_{b}\varphi^{0}c_{r}}{\alpha_{c}B^{0}\Delta \; t} )( {p_{i}^{n - 1} - p_{i}^{n}} )} + \frac{V_{b}\Delta \; ɛ_{V}}{\alpha_{c}B^{0}\Delta \; t}} < 0} & \;\end{matrix}$ production wells should be used, being the solutionapproximated of the loose flow differential equation by finitedifferences, the underlined term represents the flow rate added orremoved, as represented by the equation:${{T_{i + \frac{1}{2}}^{n}p_{i + 1}^{n + 1}} - {\lbrack {( \frac{V_{b}\varphi \; c_{f}}{a_{c}B^{0}\Delta \; t} )_{i} + ( \frac{V_{b}\varphi^{0}c_{r}}{a_{c}B\; \Delta \; t} )_{i} + T_{i + \frac{1}{2}}^{n} + T_{i - \frac{1}{2}}^{n}} \rbrack p_{i}^{n + 1}} + {T_{i - \frac{1}{2}}^{n}p_{i - 1}^{n + 1}}} = {{{- \lbrack {( \frac{V_{b}\varphi \; c_{f}}{\alpha_{c}B^{0}\Delta \; t} ) + ( \frac{V_{b}\varphi^{0}c_{r}}{a_{c}B^{0}\Delta \; t} )} \rbrack}p_{i}^{n}} - {\underset{\_}{{( \frac{V_{b}\varphi^{0}c_{r}}{\alpha_{c}B^{0}\Delta \; t} )( {p_{i}^{n - 1} - p_{i}^{n}} )} + \frac{V_{b}{\Delta ɛ}_{V}}{\alpha_{c}B^{0}\Delta \; t}}.}}$4. The method according to claim 1, wherein the adjustment of the flowequation of conventional reservoirs simulator, so that this is similarto the flow equation of the full coupling system, is driven byintroducing a pseudo-compressibility according to the followingequation:$c_{r_{pseudo}} = \frac{ɛ_{V_{i}}^{n + 1} - ɛ_{V_{i}}^{n}}{\varphi^{n + 1}( {p_{i}^{n - 1} - p_{i}^{n}} )}$in the equation approximated by finite differences of the loosedifferential flow equation, as the following equation:${{T_{i + \frac{1}{2}}^{n}p_{i + 1}^{n + 1}} - {\lbrack {( \frac{V_{b}\varphi \; c_{f}}{a_{c}B^{0}\Delta \; t} )_{i} + ( \frac{V_{b}\varphi^{0}c_{r}}{a_{c}B\; \Delta \; t} )_{i} + T_{i + \frac{1}{2}}^{n} + T_{i - \frac{1}{2}}^{n}} \rbrack p_{i}^{n + 1}} + {T_{i - \frac{1}{2}}^{n}p_{i - 1}^{n + 1}}} = {{- \lbrack {( \frac{V_{b}\varphi \; c_{f}}{a_{c}B^{0}\Delta \; t} )_{i} + ( \frac{V_{b}\varphi^{0}\; c_{r}}{a_{c}B\; \Delta \; t} )_{i}} \rbrack}{p_{i}^{n}.}}$5. A system for approximating the flow equation, comprising a computerreadable medium containing a program code means and means for performinga method according to claim
 1. 6. The method according to claim 2,wherein the said adjustment of the flow equation of the conventionalreservoir simulator, so that this is similar to the flow equation of thefull coupling system, is conducted through the addition or removal offlow rate using wells, according to equations i) or ii): $\begin{matrix} i ) & \; \\{{{{{if}\mspace{14mu} ( \frac{V_{b}\varphi^{0}c_{r}}{\alpha_{c}B^{0}\Delta \; t} )( {p_{i}^{n - 1} - p_{i}^{n}} )} + \frac{V_{b}{\Delta ɛ}_{V}}{\alpha_{c}B^{0}\Delta \; t}} > 0}{{{injection}\mspace{14mu} {wells}\mspace{14mu} {should}\mspace{14mu} {be}\mspace{14mu} {used}};{and}}} & \; \\ {ii} ) & \; \\{{{{if}\mspace{14mu} ( \frac{V_{b}\varphi^{0}c_{r}}{\alpha_{c}B^{0}\Delta \; t} )( {p_{i}^{n - 1} - p_{i}^{n}} )} + \frac{V_{b}\Delta \; ɛ_{V}}{\alpha_{c}B^{0}\Delta \; t}} < 0} & \;\end{matrix}$ production wells should be used, being the solutionapproximated of the loose flow differential equation by finitedifferences, the underlined term represents the flow rate added orremoved, as represented by the equation:${{T_{i + \frac{1}{2}}^{n}p_{i + 1}^{n + 1}} - {\lbrack {( \frac{V_{b}\varphi \; c_{f}}{a_{c}B^{0}\Delta \; t} )_{i} + ( \frac{V_{b}\varphi^{0}c_{r}}{a_{c}B\; \Delta \; t} )_{i} + T_{i + \frac{1}{2}}^{n} + T_{i - \frac{1}{2}}^{n}} \rbrack p_{i}^{n + 1}} + {T_{i - \frac{1}{2}}^{n}p_{i - 1}^{n + 1}}} = {{{- \lbrack {( \frac{V_{b}\varphi \; c_{f}}{\alpha_{c}B^{0}\Delta \; t} ) + ( \frac{V_{b}\varphi^{0}c_{r}}{\alpha_{c}B^{0}\Delta \; t} )} \rbrack}p_{i}^{n}} - {\underset{\_}{{( \frac{V_{b}\varphi^{0}c_{r}}{\alpha_{c}B^{0}\Delta \; t} )( {p_{i}^{n - 1} - p_{i}^{n}} )} + \frac{V_{b}{\Delta ɛ}_{V}}{\alpha_{c}B^{0}\Delta \; t}}.}}$7. The method according to claim 2, wherein the adjustment of the flowequation of conventional reservoirs simulator, so that this is similarto the flow equation of the full coupling system, is driven byintroducing a pseudo-compressibility according to the followingequation:$c_{r_{pseudo}} = \frac{ɛ_{V_{i}}^{n + 1} - ɛ_{V_{i}}^{n}}{\varphi^{n + 1}( {p_{i}^{n - 1} - p_{i}^{n}} )}$in the equation approximated by finite differences of the loosedifferential flow equation, as the following equation:${{T_{i + \frac{1}{2}}^{n}p_{i + 1}^{n + 1}} - {\lbrack {( \frac{V_{b}\varphi \; c_{f}}{a_{c}B^{0}\Delta \; t} )_{i} + ( \frac{V_{b}\varphi^{0}c_{r}}{a_{c}B\; \Delta \; t} )_{i} + T_{i + \frac{1}{2}}^{n} + T_{i - \frac{1}{2}}^{n}} \rbrack p_{i}^{n + 1}} + {T_{i - \frac{1}{2}}^{n}p_{i - 1}^{n + 1}}} = {{- \lbrack {( \frac{V_{b}\varphi \; c_{f}}{a_{c}B^{0}\Delta \; t} )_{i} + ( \frac{V_{b}\varphi^{0}\; c_{r}}{a_{c}B\; \Delta \; t} )_{i}} \rbrack}{p_{i}^{n}.}}$